arXiv subjects

Moritz Jirak

Publications and source records attributed to Moritz Jirak.

21 records · Page 2Linked to original sources

arXiv2015

Volatility estimation under one-sided errors with applications to limit order books

For a semi-martingale $X_t$, which forms a stochastic boundary, a rate-optimal estimator for its quadratic variation $\langle X, X \rangle_t$ is constructed based on observations in the vicinity of $X_t$. The problem is embedded in a Poisson point process framework, which reveals an interesting connection to the theory of Brownian excursion areas. We derive $n^{-1/3}$ as optimal convergence rate in a high-frequency framework with $n$ observations (in mean). We discuss a potential application for the estimation of the integrated squared volatility of an efficient price process $X_t$ from intra-day order book quotes.

math.PR↗

arXiv2014

Adaptive function estimation in nonparametric regression with one-sided errors

We consider the model of nonregular nonparametric regression where smoothness constraints are imposed on the regression function $f$ and the regression errors are assumed to decay with some sharpness level at their endpoints. The aim of this paper is to construct an adaptive estimator for the regression function $f$. In contrast to the standard model where local averaging is fruitful, the nonregular conditions require a substantial different treatment based on local extreme values. We study this model under the realistic setting in which both the smoothness degree $β>0$ and the sharpness degree $\mathfrak {a}\in(0,\infty)$ are unknown in advance. We construct adaptation procedures applying a nested version of Lepski's method and the negative Hill estimator which show no loss in the convergence rates with respect to the general $L_q$-risk and a logarithmic loss with respect to the pointwise risk. Optimality of these rates is proved for $\mathfrak{a}\in(0,\infty)$. Some numerical simulations and an application to real data are provided.

math.ST↗

arXiv2012

Simultaneous confidence bands for Yule-Walker estimators and order selection

Let $\{X_k,k\in{\mathbb{Z}}\}$ be an autoregressive process of order $q$. Various estimators for the order $q$ and the parameters ${\bolds Θ}_q=(θ_1,...,θ_q)^T$ are known; the order is usually determined with Akaike's criterion or related modifications, whereas Yule-Walker, Burger or maximum likelihood estimators are used for the parameters ${\boldsΘ}_q$. In this paper, we establish simultaneous confidence bands for the Yule--Walker estimators $\hatθ_i$; more precisely, it is shown that the limiting distribution of ${\max_{1\leq i\leq d_n}}|\hatθ_i-θ_i|$ is the Gumbel-type distribution $e^{-e^{-z}}$, where $q\in\{0,...,d_n\}$ and $d_n=\mathcal {O}(n^δ)$, $δ>0$. This allows to modify some of the currently used criteria (AIC, BIC, HQC, SIC), but also yields a new class of consistent estimators for the order $q$. These estimators seem to have some potential, since they outperform most of the previously mentioned criteria in a small simulation study. In particular, if some of the parameters $\{θ_i\}_{1\leq i\leq d_n}$ are zero or close to zero, a significant improvement can be observed. As a byproduct, it is shown that BIC, HQC and SIC are consistent for $q\in\{0,...,d_n\}$ where $d_n=\mathcal {O}(n^δ)$.

math.ST↗